R-Ladies Philly
2024-12-17
Because time-to-event data are common in many fields, it also goes by names besides survival analysis including:
Censoring occurs when the event of interest is not observed after a period of follow-up
Binary data doesn’t have the ability to change depending on the time of analysis, e.g. 5-year survival will have the same value whether it is analyzed at 5 years and 1 day, 5 years and 2 days, 6 years, etc. Either a participant died by 5 years or they didn’t.
Time-to-event data may have different values depending on the time of analysis, e.g. overall survival will have different values depending on whether it is analyzed at 5 years and 1 day or at 6 years, since additional participants can die between those two time points.
A subject may be censored due to:
Today we will focus only on right censoring.
How would we compute the proportion who are event-free at 18 years?
To analyze survival data, we need to know the observed time \(Y_i\) and the event indicator \(\delta_i\). For subject \(i\):
The probability that a subject will survive beyond any given specified time
\[S(t) = Pr(T>t) = 1 - F(t)\]
\(S(t)\): survival function \(F(t) = Pr(T \leq t)\): cumulative distribution function
In theory the survival function is smooth; in practice we observe events on a discrete time scale.
The survival probability at a certain time, \(S(t)\), is a conditional probability of surviving beyond that time, given that an individual has survived just prior to that time. The survival probability can be estimated as the number of patients who are alive without loss to follow-up at that time, divided by the number of patients who were alive just prior to that time.
The Kaplan-Meier estimate of survival probability at a given time is the product of these conditional probabilities up until that given time.
At time 0, the survival probability is 1, i.e. \(S(t_0) = 1\).
NEED TO ADD DATES TO THE EXAMPLE DATA NEXT
To access the example data used throughout this talk, install and load the {cancersimdata} package from my GitHub repo:
Clark, T., Bradburn, M., Love, S., & Altman, D. (2003). Survival analysis part I: Basic concepts and first analyses. 232-238. ISSN 0007-0920.
M J Bradburn, T G Clark, S B Love, & D G Altman. (2003). Survival Analysis Part II: Multivariate data analysis – an introduction to concepts and methods. British Journal of Cancer, 89(3), 431-436.
Bradburn, M., Clark, T., Love, S., & Altman, D. (2003). Survival analysis Part III: Multivariate data analysis – choosing a model and assessing its adequacy and fit. 89(4), 605-11.
Clark, T., Bradburn, M., Love, S., & Altman, D. (2003). Survival analysis part IV: Further concepts and methods in survival analysis. 781-786. ISSN 0007-0920.